L9a49

L9a48

L9a50

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a49 at Knotilus!
	L9a49 is $9_{6}^{3}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a49's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X16,10,17,9 X14,8,15,7 X18,14,11,13 X10,16,5,15 X8,18,9,17 X2536 X4,11,1,12
Gauss code	{1, -8, 2, -9}, {8, -1, 4, -7, 3, -6}, {9, -2, 5, -4, 6, -3, 7, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv^{2}w-uv^{2}+uvw^{2}-3uvw+2uv-uw^{2}+2uw-2v^{2}w+v^{2}-2vw^{2}+3vw-v+w^{2}-w}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$-q^{5}+3q^{4}+q^{-4}-4q^{3}-2q^{-3}+7q^{2}+5q^{-2}-7q-6q^{-1}+8$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{2}a^{-4}+a^{4}+z^{4}a^{-2}-2a^{2}z^{2}+a^{2}z^{-2}+z^{2}a^{-2}+a^{-2}z^{-2}+2a^{-2}+z^{4}-z^{2}-2z^{-2}-3$ (db)
Kauffman polynomial	$z^{8}a^{-2}+z^{8}+2az^{7}+5z^{7}a^{-1}+3z^{7}a^{-3}+3a^{2}z^{6}+4z^{6}a^{-2}+3z^{6}a^{-4}+4z^{6}+2a^{3}z^{5}+2az^{5}-7z^{5}a^{-1}-6z^{5}a^{-3}+z^{5}a^{-5}+a^{4}z^{4}-4a^{2}z^{4}-13z^{4}a^{-2}-8z^{4}a^{-4}-10z^{4}-2a^{3}z^{3}-7az^{3}-z^{3}a^{-1}+2z^{3}a^{-3}-2z^{3}a^{-5}-2a^{4}z^{2}+4a^{2}z^{2}+11z^{2}a^{-2}+5z^{2}a^{-4}+12z^{2}+6az+6za^{-1}+a^{4}-3a^{2}-5a^{-2}-8-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 11                   1 -1 9                 2   2 7               2 1   -1 5             5 2     3 3           4 4       0 1         4 3         1 -1       3 5           2 -3     2 3             -1 -5     3               3 -7 1 2                 -1 -9 1                   1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-4$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{4}$ $r=1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{5}$ $r=3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

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